Math 1508
Rational Function Overview
A rational function has the form f  x  
N  x  an x n  an1 x n 1  ...  a2 x 2  a1 x  a0
. Frequently, it

D  x  bm x m  bm1 x m1  ...  b2 x 2  b1 x  b0
will be a good idea to completely factor the numerator and denominator.
Domain – The domain is found by setting the denominator not equal to zero and solving. That is,
D  x  0 .
Vertical Asymptotes – Vertical asymptotes are vertical lines where the function “blows up.” These occur
when the denominator is zero but the numerator is not. That is, D  x   0, N  x   0 .
Holes – Holes appear in the graph when you have an x-value that can be factored from both the
numerator and the denominator. That is, holes appear when both D  x   0, N  x   0 .
x-intercepts – These intercepts occur when the y-value is zero. If you have a rational function equal to
zero, your first step is to multiply both sides by the denominator. In so doing, you end up solving the
numerator equal to zero. That is, x-intercepts occur when N  x   0, D  x   0 .
y-intercepts – These intercepts occur when the x-value is zero. Knowing this, we can substitute zero in

for all the x’s and find that the y-intercept, if it exists, is the point  0,

a0 
 . Notice that if the
b0 
denominator does not have a constant term, the graph will not have a y-intercept.
Horizontal Asymptotes – Horizontal asymptotes are horizontal lines where the function “eventually”
settles. We sometimes refer to this as the end behavior of the function as it is what happens to f as x
goes to either infinity.
a) If n < m, y = 0 is horizontal asymptote.
b) If n = m, y 
an
is horizontal asymptote. I call these the leading coefficients, assuming the
bm
polynomials are written in standard descending order.
c) If n > m, there is no horizontal asymptote (but there may be a slant asymptote).
Slant (Oblique) Asymptotes – Slant asymptotes are linear equations. Slant asymptotes exist if the
relation to the degrees of the numerator and denominator are exactly n = m + 1. That is, the numerator
is exactly one degree higher than the denominator. To find the slant asymptote, divide the polynomial
to write it in proper form. Ignoring the remainder, your asymptote is the linear function that is the
quotient.